IODS course project
Mohan Babu
- About the project
- the data has 166 observations and 7 variables
- Mostly likert scale (1-5) variables
- Includes also Age (positive integers) and gender (as a two level factor: F and M)
- Attitude = Global attitude toward statistics ~Da+Db+Dc+Dd+De+Df+Dg+Dh+Di+Dj
- Points=
- Regression and model validation
- the data has 166 observations and 7 variables
- Mostly likert scale (1-5) variables
- Includes also Age (positive integers) and gender (as a two level factor: F and M)
- Attitude = Global attitude toward statistics ~Da+Db+Dc+Dd+De+Df+Dg+Dh+Di+Dj
- Points=Exam points (0-33)
- ## Reading data
- There were 382 observations and 35 variables. Two datasets are provided regarding the performance in two distinct subjects: Mathematics (mat) and Portuguese language (por) were joned using variables “school”, “sex”, “age”, “address”, “famsize”, “Pstatus”, “Medu”, “Fedu”, “Mjob”, “Fjob”, “reason”, “nursery”,“internet” as (student) identifiers. The relation between variables in joined data and its relation with aclohol consumption was evaluated
- # access the tidyverse libraries tidyr, dplyr, ggplot2
- high alcohol consumption seem to correlate with poor grades in both sexes
- High alcohol consumption seems to be correlated with higher absences in both sexes
- Student freetime seems not to influence alcohol consumption
- Age of the students seems not to corrlate with high alcohol consumptipn
- For every 1 point increase in absence, the risk of high alcohol consumption (OR) is 0.07294 and is highly singificant.
- for every 1 point increase in failure, the risk of high alcohol consumption (OR) is 0.40462 and is highly singificant
- compared to females, males have higher alcohol consumption
- For every 1 point increase in failures, the risk of high aclhol consumption is 1.4987270
- For every 1 point increase in absences, the risk of high aclhol consumption is 1.0756623
- Every 4th model predictions on high alcohol consumption are wrong
- 1. Loading the Boston data from MASS package
- 3. Graphical overview of Boston data
- 3. correlation matrix of Boston data
- 4. print correlation matrix
- 4. visualize correlation matrix
- 4. Scaling the dataset and summary of the scaled data
- 4. creating quantile vector of crime rate
- 4. creating categorical variable of crime rate
- 4. Dropping old crime data
- 4. Adding the new categorical value to scaled data
- 4. Dividing the dataset to train and test sets
- 4. Dividing the dataset to train and test sets to include 80% of the data to the train set
- 4.creating train set
- 4. Creating test set
- 4. Saving the correct classes from the test data
- 4. remove the crime variable from the test data
- 5. Fitting the linear discriminant analysis on the train set
- 5. Drawing the LDA (bi)plot
- 6. Fitting the linear discriminant analysis on the test set
- 7,reloading the Boston data and scaling
- 7. k-means algorithm on the dataset and investigating optimal number of clusters
- Bonus exercise
- performing k-means on original Boston data
- Fitting the linear discriminant analysis on the original Boston data
- Drawing the LDA (bi)plot
- Super-bonus exercise
- checking the dimensions
- check the dimensions
- matrix multiplication
- 5 reading the human data
- 5.1 Graphical overview of the data
- 5.1 summaries of the varaibles in the human data
- 5.2 Prinicipal component analysis on the non-standardised human data
- 5.2 Biplot displying the observations by the first two principal components
- 5.3 Standardizing the variables in the human data
- 5.3 Biplot displying the observations by the first two principal components in the standardized data
- 5.4 Biplot displying the observations by the first two principal components in the standardized data
- 5.5 Tea data analysis
- 5.5 Multiple correspondence analysis on the tea data
- PART I
- 6.1.1 Reading RATS data
- 6.1.2 Factoring the categorical variables in RATS data
- 6.1.3 Converting wide form to long form of RATS data
- 6.1.2 Plotting the RATSL data
- 6.1.5 Plotting the RATSL data
- 6.1.7 Plotting again with the standardised Weight
- 6.1.8 Number of WD, Summary data with mean and standard error of Weight by group and time
- 6.1.9 Plotting the mean profiles of RATSS data
- 6.1.10 Creating a summary data by Group and ID with mean as the summary variable (ignoring baseline WD 0).
- 6.1.11 Drawing a boxplot of the mean weights versus Group
- 6.1.12 Creating a new RATS data by filtering the outlier
- 6.1.13 Adjusting the ggplot code and drawing the plot again with the new data
- 6.1.14 Adding the baseline from the original data as a new variable to the summary data
- 6.1.15 Fitting the linear model with the mean as the response
- 6.1.16 Computing the analysis of variance table for the fitted model with anova()
- PART II
- 6.2.1 Reading the BPRS data
- 6.2.2 Factoring the categorical variables in BPRS data
- 6.2.3 Converting wide form to long form of BPRS data
- 6.2.4 Plotting the BPRSL data
- 6.2.5 Creating a regression model BPRS_reg
- 6.2.6 Creating a random intercept model for BPRSL data
- 6.2.7 creating a random intercept and random slope model for BPRSL data
- 6.2.8 Performing an ANOVA test on the two models of BPRS data
- 6.2.9 Creating a random intercept and random slope model
- 6.2.10 Perform an ANOVA test on the two models
- 6.2.11 Drawing the plot of BPRSL
- 6.2.12 Creating a vector of the fitted values
- 6.2.13 Creating a new column fitted to BPRSL data
- 6.2.14 Drawing the plot of BPRSL
About the project
Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you can use R Markdown syntax. ‘I am feeling great’ ‘I heard of this course from University of Eastern Finland Pages’ ‘I expect to use R program and datascience tools for my research’
Here is the link to my Girhub repository: https://github.com/mohanbabu29/IODS-project
and Here is the link to the my [diary page]: (https://mohanbabu29.github.io/IODS-project/).
<<<<<<< HEAD
output: html_document: default pdf_document: default — # Regression and model validation
Describe the work you have done this week and summarize your learning.
- Describe your work and results clearly.
- Assume the reader has an introductory course level understanding of writing and reading R code as well as statistical methods.
- Assume the reader has no previous knowledge of your data or the more advanced methods you are using.
This week we understood data wrangling, perform explanatory examination and fit a simple linear model to the data.
Reading data
Let’s read the data
library(dplyr)
learning2014 <-readxl::read_excel("~/IODS-project/data 2/learning2014.xlsx") %>%
mutate_at(vars(gender), factor)str(learning2014)## Classes 'tbl_df', 'tbl' and 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : num 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : num 25 12 24 10 22 21 21 31 24 26 ...the data has 166 observations and 7 variables
Mostly likert scale (1-5) variables
Includes also Age (positive integers) and gender (as a two level factor: F and M)
Attitude = Global attitude toward statistics ~Da+Db+Dc+Dd+De+Df+Dg+Dh+Di+Dj
Points=
library(ggplot2)
pairs(learning2014[!names(learning2014) %in% c("gender")],col=learning2014$gender)
summary(learning2014)## gender age attitude deep stra
## F:110 Min. :17.00 Min. :1.400 Min. :1.583 Min. :1.250
## M: 56 1st Qu.:21.00 1st Qu.:2.600 1st Qu.:3.333 1st Qu.:2.625
## Median :22.00 Median :3.200 Median :3.667 Median :3.188
## Mean :25.51 Mean :3.143 Mean :3.680 Mean :3.121
## 3rd Qu.:27.00 3rd Qu.:3.700 3rd Qu.:4.083 3rd Qu.:3.625
## Max. :55.00 Max. :5.000 Max. :4.917 Max. :5.000
## surf points
## Min. :1.583 Min. : 7.00
## 1st Qu.:2.417 1st Qu.:19.00
## Median :2.833 Median :23.00
## Mean :2.787 Mean :22.72
## 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :4.333 Max. :33.00library(GGally)
library(ggplot2)
# create a more advanced plot matrix with ggpairs()
ggpairs(learning2014,
mapping = aes(col = gender, alpha = 0.3),
lower = list(combo = wrap("facethist", bins = 20))
)
qplot(attitude, points, data = learning2014) + geom_smooth(method = "lm")
my_model <- lm(points ~ attitude, data = learning2014)
results <- summary(my_model)
knitr::kable(results$coefficients, digits=3, caption="Regression coefficients")| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 11.637 | 1.830 | 6.358 | 0 |
| attitude | 3.525 | 0.567 | 6.214 | 0 |
plot(my_model, which=c(1,2,5))Regression and model validation
Describe the work you have done this week and summarize your learning.
- Describe your work and results clearly.
- Assume the reader has an introductory course level understanding of writing and reading R code as well as statistical methods.
- Assume the reader has no previous knowledge of your data or the more advanced methods you are using.
This week we understood data wrangling, perform explanatory examination and fit a simple linear model to the data.
Reading data
Let’s read the data
library(dplyr)
learning2014 <-readxl::read_excel("~/IODS-project/data 2/learning2014.xlsx") %>%
mutate_at(vars(gender), factor)str(learning2014)## Classes 'tbl_df', 'tbl' and 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : num 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : num 25 12 24 10 22 21 21 31 24 26 ...the data has 166 observations and 7 variables
Mostly likert scale (1-5) variables
Includes also Age (positive integers) and gender (as a two level factor: F and M)
Attitude = Global attitude toward statistics ~Da+Db+Dc+Dd+De+Df+Dg+Dh+Di+Dj
Points=Exam points (0-33)
dim(learning2014)## [1] 166 7library(ggplot2)
pairs(learning2014[!names(learning2014) %in% c("gender")],col=learning2014$gender)
summary(learning2014)## gender age attitude deep stra
## F:110 Min. :17.00 Min. :1.400 Min. :1.583 Min. :1.250
## M: 56 1st Qu.:21.00 1st Qu.:2.600 1st Qu.:3.333 1st Qu.:2.625
## Median :22.00 Median :3.200 Median :3.667 Median :3.188
## Mean :25.51 Mean :3.143 Mean :3.680 Mean :3.121
## 3rd Qu.:27.00 3rd Qu.:3.700 3rd Qu.:4.083 3rd Qu.:3.625
## Max. :55.00 Max. :5.000 Max. :4.917 Max. :5.000
## surf points
## Min. :1.583 Min. : 7.00
## 1st Qu.:2.417 1st Qu.:19.00
## Median :2.833 Median :23.00
## Mean :2.787 Mean :22.72
## 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :4.333 Max. :33.00library(GGally)
library(ggplot2)
# create a more advanced plot matrix with ggpairs()
ggpairs(learning2014,
mapping = aes(col = gender, alpha = 0.3),
lower = list(combo = wrap("facethist", bins = 20))
)
qplot(attitude, points, data = learning2014) + geom_smooth(method = "lm")
my_model <- lm(points ~ attitude, data = learning2014)
results <- summary(my_model)
knitr::kable(results$coefficients, digits=3, caption="Regression coefficients")| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 11.637 | 1.830 | 6.358 | 0 |
| attitude | 3.525 | 0.567 | 6.214 | 0 |
plot(my_model, which=c(1,2,5)) >>>>>>> c98635c5e8506d39b39832ad4d622771b7ae5bed “C:/Program Files/Git/bin/git” config –mohanbabu29 ***
## Reading data
Let’s read the data
library(dplyr)
alc<- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/alc.txt", sep=",", header=TRUE)print<-vars(alc)str(alc)## 'data.frame': 382 obs. of 35 variables:
## $ school : Factor w/ 2 levels "GP","MS": 1 1 1 1 1 1 1 1 1 1 ...
## $ sex : Factor w/ 2 levels "F","M": 1 1 1 1 1 2 2 1 2 2 ...
## $ age : int 18 17 15 15 16 16 16 17 15 15 ...
## $ address : Factor w/ 2 levels "R","U": 2 2 2 2 2 2 2 2 2 2 ...
## $ famsize : Factor w/ 2 levels "GT3","LE3": 1 1 2 1 1 2 2 1 2 1 ...
## $ Pstatus : Factor w/ 2 levels "A","T": 1 2 2 2 2 2 2 1 1 2 ...
## $ Medu : int 4 1 1 4 3 4 2 4 3 3 ...
## $ Fedu : int 4 1 1 2 3 3 2 4 2 4 ...
## $ Mjob : Factor w/ 5 levels "at_home","health",..: 1 1 1 2 3 4 3 3 4 3 ...
## $ Fjob : Factor w/ 5 levels "at_home","health",..: 5 3 3 4 3 3 3 5 3 3 ...
## $ reason : Factor w/ 4 levels "course","home",..: 1 1 3 2 2 4 2 2 2 2 ...
## $ nursery : Factor w/ 2 levels "no","yes": 2 1 2 2 2 2 2 2 2 2 ...
## $ internet : Factor w/ 2 levels "no","yes": 1 2 2 2 1 2 2 1 2 2 ...
## $ guardian : Factor w/ 3 levels "father","mother",..: 2 1 2 2 1 2 2 2 2 2 ...
## $ traveltime: int 2 1 1 1 1 1 1 2 1 1 ...
## $ studytime : int 2 2 2 3 2 2 2 2 2 2 ...
## $ failures : int 0 0 3 0 0 0 0 0 0 0 ...
## $ schoolsup : Factor w/ 2 levels "no","yes": 2 1 2 1 1 1 1 2 1 1 ...
## $ famsup : Factor w/ 2 levels "no","yes": 1 2 1 2 2 2 1 2 2 2 ...
## $ paid : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 1 1 2 2 ...
## $ activities: Factor w/ 2 levels "no","yes": 1 1 1 2 1 2 1 1 1 2 ...
## $ higher : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
## $ romantic : Factor w/ 2 levels "no","yes": 1 1 1 2 1 1 1 1 1 1 ...
## $ famrel : int 4 5 4 3 4 5 4 4 4 5 ...
## $ freetime : int 3 3 3 2 3 4 4 1 2 5 ...
## $ goout : int 4 3 2 2 2 2 4 4 2 1 ...
## $ Dalc : int 1 1 2 1 1 1 1 1 1 1 ...
## $ Walc : int 1 1 3 1 2 2 1 1 1 1 ...
## $ health : int 3 3 3 5 5 5 3 1 1 5 ...
## $ absences : int 6 4 10 2 4 10 0 6 0 0 ...
## $ G1 : int 5 5 7 15 6 15 12 6 16 14 ...
## $ G2 : int 6 5 8 14 10 15 12 5 18 15 ...
## $ G3 : int 6 6 10 15 10 15 11 6 19 15 ...
## $ alc_use : num 1 1 2.5 1 1.5 1.5 1 1 1 1 ...
## $ high_use : logi FALSE FALSE TRUE FALSE FALSE FALSE ...There were 382 observations and 35 variables. Two datasets are provided regarding the performance in two distinct subjects: Mathematics (mat) and Portuguese language (por) were joned using variables “school”, “sex”, “age”, “address”, “famsize”, “Pstatus”, “Medu”, “Fedu”, “Mjob”, “Fjob”, “reason”, “nursery”,“internet” as (student) identifiers. The relation between variables in joined data and its relation with aclohol consumption was evaluated
# access the tidyverse libraries tidyr, dplyr, ggplot2
alc <- mutate(alc, high_use = alc_use > 2)glimpse(alc)## Observations: 382
## Variables: 35
## $ school <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP,...
## $ sex <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, ...
## $ age <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15,...
## $ address <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, ...
## $ famsize <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, G...
## $ Pstatus <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, ...
## $ Medu <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, ...
## $ Fedu <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, ...
## $ Mjob <fct> at_home, at_home, at_home, health, other, services,...
## $ Fjob <fct> teacher, other, other, services, other, other, othe...
## $ reason <fct> course, course, other, home, home, reputation, home...
## $ nursery <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, ye...
## $ internet <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes,...
## $ guardian <fct> mother, father, mother, mother, father, mother, mot...
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, ...
## $ studytime <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, ...
## $ failures <int> 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ schoolsup <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, ...
## $ famsup <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes,...
## $ paid <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, ...
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes,...
## $ higher <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, y...
## $ romantic <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no...
## $ famrel <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, ...
## $ freetime <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, ...
## $ goout <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, ...
## $ Dalc <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
## $ Walc <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, ...
## $ health <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, ...
## $ absences <int> 6, 4, 10, 2, 4, 10, 0, 6, 0, 0, 0, 4, 2, 2, 0, 4, 6...
## $ G1 <int> 5, 5, 7, 15, 6, 15, 12, 6, 16, 14, 10, 10, 14, 10, ...
## $ G2 <int> 6, 5, 8, 14, 10, 15, 12, 5, 18, 15, 8, 12, 14, 10, ...
## $ G3 <int> 6, 6, 10, 15, 10, 15, 11, 6, 19, 15, 9, 12, 14, 11,...
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1...
## $ high_use <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FAL...alc %>% group_by(alc_use,age) %>% summarise(count = n())## # A tibble: 41 x 3
## # Groups: alc_use [9]
## alc_use age count
## <dbl> <int> <int>
## 1 1 15 46
## 2 1 16 41
## 3 1 17 26
## 4 1 18 27
## 5 1 19 3
## 6 1 20 1
## 7 1.5 15 9
## 8 1.5 16 24
## 9 1.5 17 20
## 10 1.5 18 12
## # ... with 31 more rowsalc %>% group_by(alc_use,sex) %>% summarise(count = n())## # A tibble: 17 x 3
## # Groups: alc_use [9]
## alc_use sex count
## <dbl> <fct> <int>
## 1 1 F 89
## 2 1 M 55
## 3 1.5 F 41
## 4 1.5 M 27
## 5 2 F 27
## 6 2 M 31
## 7 2.5 F 25
## 8 2.5 M 17
## 9 3 F 11
## 10 3 M 21
## 11 3.5 F 3
## 12 3.5 M 14
## 13 4 F 1
## 14 4 M 8
## 15 4.5 M 3
## 16 5 F 1
## 17 5 M 8alc %>% group_by(alc_use,Medu) %>% summarise(count = n())## # A tibble: 37 x 3
## # Groups: alc_use [9]
## alc_use Medu count
## <dbl> <int> <int>
## 1 1 0 1
## 2 1 1 18
## 3 1 2 41
## 4 1 3 33
## 5 1 4 51
## 6 1.5 1 10
## 7 1.5 2 22
## 8 1.5 3 12
## 9 1.5 4 24
## 10 2 1 5
## # ... with 27 more rowsalc %>% group_by(alc_use,Fedu) %>% summarise(count = n())## # A tibble: 35 x 3
## # Groups: alc_use [9]
## alc_use Fedu count
## <dbl> <int> <int>
## 1 1 0 2
## 2 1 1 28
## 3 1 2 42
## 4 1 3 38
## 5 1 4 34
## 6 1.5 1 16
## 7 1.5 2 15
## 8 1.5 3 19
## 9 1.5 4 18
## 10 2 1 9
## # ... with 25 more rowsalc %>% group_by(alc_use,age) %>% boxplot
high alcohol consumption seem to correlate with poor grades in both sexes
Student freetime seems not to influence alcohol consumption
Age of the students seems not to corrlate with high alcohol consumptipn
m <- glm(high_use ~ failures + absences + sex, data = alc, family = "binomial")summary(m)##
## Call:
## glm(formula = high_use ~ failures + absences + sex, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.6629 -0.8545 -0.5894 1.0339 2.0428
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.95397 0.22819 -8.563 < 2e-16 ***
## failures 0.40462 0.15024 2.693 0.00708 **
## absences 0.07294 0.01796 4.061 4.88e-05 ***
## sexM 0.98848 0.24453 4.042 5.29e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 462.21 on 381 degrees of freedom
## Residual deviance: 418.64 on 378 degrees of freedom
## AIC: 426.64
##
## Number of Fisher Scoring iterations: 4coef(m)## (Intercept) failures absences sexM
## -1.95396790 0.40461608 0.07293654 0.98847614OR <- coef(m) %>% expCI <- confint(m) %>% exp## Waiting for profiling to be done...cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.1417107 0.08883883 0.2178283
## failures 1.4987270 1.11549818 2.0187171
## absences 1.0756623 1.04072883 1.1163576
## sexM 2.6871365 1.67434331 4.3755694For every 1 point increase in absence, the risk of high alcohol consumption (OR) is 0.07294 and is highly singificant.
for every 1 point increase in failure, the risk of high alcohol consumption (OR) is 0.40462 and is highly singificant
compared to females, males have higher alcohol consumption
For every 1 point increase in failures, the risk of high aclhol consumption is 1.4987270
For every 1 point increase in absences, the risk of high aclhol consumption is 1.0756623
probabilities <- predict(m, type = "response")alc <- mutate(alc, probability = probabilities)alc <- mutate(alc, prediction = probability > 0.5)select(alc, failures, absences, sex, high_use, probability, prediction) %>% tail(10)## failures absences sex high_use probability prediction
## 373 1 0 M FALSE 0.3633449 FALSE
## 374 1 14 M TRUE 0.6130701 TRUE
## 375 0 2 F FALSE 0.1408685 FALSE
## 376 0 7 F FALSE 0.1910175 FALSE
## 377 1 0 F FALSE 0.1751799 FALSE
## 378 0 0 F FALSE 0.1241213 FALSE
## 379 1 0 F FALSE 0.1751799 FALSE
## 380 1 0 F FALSE 0.1751799 FALSE
## 381 0 3 M TRUE 0.3215447 FALSE
## 382 0 0 M TRUE 0.2757800 FALSEtable(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 258 12
## TRUE 86 26table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins## prediction
## high_use FALSE TRUE Sum
## FALSE 0.67539267 0.03141361 0.70680628
## TRUE 0.22513089 0.06806283 0.29319372
## Sum 0.90052356 0.09947644 1.00000000loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2565445loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)}loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2565445Every 4th model predictions on high alcohol consumption are wrong
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)cv$delta[1]## [1] 0.2565445# chapter 4 exercise, Mohan Babu, 25.11.2019
title: “Chapter4” output: html_document —
1. Loading the Boston data from MASS package
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectstr(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14==================================================== # The Boston data consists of housing values in suburbs of Boston with data frame consisting of 506 rows and 14 columns containing variables. ========================================================
3. Graphical overview of Boston data
library(ggplot2)
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
3. correlation matrix of Boston data
cor_matrix<-cor(Boston)4. print correlation matrix
print(cor_matrix)## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593
## ptratio black lstat medv
## crim 0.2899456 -0.38506394 0.4556215 -0.3883046
## zn -0.3916785 0.17552032 -0.4129946 0.3604453
## indus 0.3832476 -0.35697654 0.6037997 -0.4837252
## chas -0.1215152 0.04878848 -0.0539293 0.1752602
## nox 0.1889327 -0.38005064 0.5908789 -0.4273208
## rm -0.3555015 0.12806864 -0.6138083 0.6953599
## age 0.2615150 -0.27353398 0.6023385 -0.3769546
## dis -0.2324705 0.29151167 -0.4969958 0.2499287
## rad 0.4647412 -0.44441282 0.4886763 -0.3816262
## tax 0.4608530 -0.44180801 0.5439934 -0.4685359
## ptratio 1.0000000 -0.17738330 0.3740443 -0.5077867
## black -0.1773833 1.00000000 -0.3660869 0.3334608
## lstat 0.3740443 -0.36608690 1.0000000 -0.7376627
## medv -0.5077867 0.33346082 -0.7376627 1.00000004. visualize correlation matrix
corrplot::corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)
====================== Crime rates are strongly correlated with index of accessibility to radial highways ======================
4. Scaling the dataset and summary of the scaled data
boston_scaled<-scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.98654. creating quantile vector of crime rate
class(boston_scaled)## [1] "matrix"boston_scaled<-as.data.frame(boston_scaled)summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110bins<-quantile(boston_scaled$crim)
print(bins)## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.9241096104. creating categorical variable of crime rate
crime<-cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label<-c("low","med_low","med_high","high"))table(crime)## crime
## low med_low med_high high
## 127 126 126 1274. Dropping old crime data
boston_scaled <-dplyr::select(boston_scaled, -crim)4. Adding the new categorical value to scaled data
boston_scaled <-data.frame(boston_scaled, crime)4. Dividing the dataset to train and test sets
boston_scaled <-data.frame(boston_scaled, crime)
n<-nrow(boston_scaled)4. Dividing the dataset to train and test sets to include 80% of the data to the train set
ind<-sample(n,size = n*0.8)4.creating train set
train <- boston_scaled[ind,]4. Creating test set
test <- boston_scaled[-ind,]4. Saving the correct classes from the test data
correct_classes<-(test$crime)4. remove the crime variable from the test data
test <- dplyr::select(test, -crime)5. Fitting the linear discriminant analysis on the train set
lda.fit <- lda(crime~zn+indus+chas+nox+rm+age+dis+rad+tax+ptratio+black+lstat+medv, data = train)
lda.fit## Call:
## lda(crime ~ zn + indus + chas + nox + rm + age + dis + rad +
## tax + ptratio + black + lstat + medv, data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2698020 0.2277228 0.2450495 0.2574257
##
## Group means:
## zn indus chas nox rm
## low 0.9290821 -0.9043974 -0.12784833 -0.8811777 0.45136424
## med_low -0.0449532 -0.3522661 -0.01556166 -0.6080292 -0.14858057
## med_high -0.4075493 0.1798342 0.20489520 0.3998634 0.07224354
## high -0.4872402 1.0170690 -0.08304540 1.0894043 -0.52326309
## age dis rad tax ptratio
## low -0.8799431 0.8757234 -0.6836912 -0.7390553 -0.4545029
## med_low -0.3811610 0.4085716 -0.5412094 -0.4611175 -0.1295800
## med_high 0.4411915 -0.3657406 -0.4435998 -0.3185150 -0.2500720
## high 0.8155168 -0.8564699 1.6386213 1.5144083 0.7813507
## black lstat medv
## low 0.37709554 -0.76762917 0.52591755
## med_low 0.31030831 -0.11933711 0.01345156
## med_high 0.05261903 0.05490616 0.13360834
## high -0.83537529 0.92160860 -0.77847650
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.08953054 0.56113425 -1.0101364
## indus 0.07218634 -0.22174594 0.2397393
## chas -0.09531663 -0.05996422 0.1169146
## nox 0.27621683 -0.79385816 -1.3189436
## rm -0.13342457 -0.11478356 -0.2125507
## age 0.22966338 -0.41609631 -0.1325351
## dis -0.02814771 -0.22710599 0.2165613
## rad 3.59273354 1.08337388 -0.2786836
## tax 0.18626877 -0.09533914 0.8325966
## ptratio 0.10340365 -0.04811054 -0.3811377
## black -0.09074088 0.06197170 0.1196124
## lstat 0.21828270 -0.22692557 0.4064369
## medv 0.19223067 -0.42219229 -0.1267976
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9580 0.0322 0.00985. Drawing the LDA (bi)plot
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}classes<-as.numeric(train$crime)plot(lda.fit, col=classes, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
6. Fitting the linear discriminant analysis on the test set
lda.pred <- predict(lda.fit, newdata = test)
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 14 3 1 0
## med_low 7 18 9 0
## med_high 1 7 16 3
## high 0 0 0 23========================================= Classifier seem to predict the crime rates correctly =========================================
7,reloading the Boston data and scaling
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00class(Boston)## [1] "data.frame"dist_eu<-(Boston)
summary(dist_eu)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.007. k-means algorithm on the dataset and investigating optimal number of clusters
set.seed(123)
k_max <- (10)
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
km<-kmeans(Boston, centers = 2)
pairs(Boston[1:5], col = km$cluster)
=========================================================================================== Optimal number of clusters seems to be 2. Crime rates seems to be highly correlated with nox ===========================================================================================
Bonus exercise
performing k-means on original Boston data
km <- kmeans(Boston, centers = 3)
pairs(Boston[1:5], col = km$cluster)
# Scaling original Boston data
boston_scaled <- scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865class(boston_scaled)## [1] "matrix"boston_scaled<-as.data.frame(boston_scaled)Fitting the linear discriminant analysis on the original Boston data
lda.fit <- lda(km$cluster~ ., data = boston_scaled)
lda.fit## Call:
## lda(km$cluster ~ ., data = boston_scaled)
##
## Prior probabilities of groups:
## 1 2 3
## 0.5296443 0.1996047 0.2707510
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3920779 0.27670879 -0.6513071 0.0214843827 -0.6152775 0.2573427
## 2 -0.3293317 -0.07332724 0.2818828 0.0005392655 0.2816899 -0.1453417
## 3 1.0097765 -0.48724019 1.0662784 -0.0424254043 0.9959393 -0.3962652
## age dis rad tax ptratio black
## 1 -0.4572006 0.5121870 -0.6013344 -0.78136288 -0.2690134 0.34109296
## 2 0.1822823 -0.2378455 -0.5418150 -0.01444889 -0.3768823 0.07010933
## 3 0.7599946 -0.8265965 1.5757732 1.53915759 0.8040926 -0.71893398
## lstat medv
## 1 -0.43621538 0.36234147
## 2 0.01371321 -0.03812375
## 3 0.84321670 -0.68070813
##
## Coefficients of linear discriminants:
## LD1 LD2
## crim 0.048210477 0.05079118
## zn 0.253528315 0.06311589
## indus 0.369497254 0.12674727
## chas -0.047064817 0.01998369
## nox -0.063156250 -0.49621758
## rm -0.005144383 0.09537352
## age -0.118710969 0.05412142
## dis -0.385151599 0.17969944
## rad 1.996321584 3.05733525
## tax 4.535785039 -2.77688761
## ptratio 0.122064688 0.19196217
## black -0.029200518 0.06353722
## lstat 0.085030308 0.12666624
## medv 0.157444662 -0.10356584
##
## Proportion of trace:
## LD1 LD2
## 0.9812 0.0188Drawing the LDA (bi)plot
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}classes<-as.numeric(km$cluster)plot(lda.fit, col=classes, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
====================================================== tax and rad are the most influential linear separators for the clusters ======================================================
Super-bonus exercise
lda.fit <- lda(crime~zn+indus+chas+nox+rm+age+dis+rad+tax+ptratio+black+lstat+medv, data = train)
lda.fit## Call:
## lda(crime ~ zn + indus + chas + nox + rm + age + dis + rad +
## tax + ptratio + black + lstat + medv, data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2698020 0.2277228 0.2450495 0.2574257
##
## Group means:
## zn indus chas nox rm
## low 0.9290821 -0.9043974 -0.12784833 -0.8811777 0.45136424
## med_low -0.0449532 -0.3522661 -0.01556166 -0.6080292 -0.14858057
## med_high -0.4075493 0.1798342 0.20489520 0.3998634 0.07224354
## high -0.4872402 1.0170690 -0.08304540 1.0894043 -0.52326309
## age dis rad tax ptratio
## low -0.8799431 0.8757234 -0.6836912 -0.7390553 -0.4545029
## med_low -0.3811610 0.4085716 -0.5412094 -0.4611175 -0.1295800
## med_high 0.4411915 -0.3657406 -0.4435998 -0.3185150 -0.2500720
## high 0.8155168 -0.8564699 1.6386213 1.5144083 0.7813507
## black lstat medv
## low 0.37709554 -0.76762917 0.52591755
## med_low 0.31030831 -0.11933711 0.01345156
## med_high 0.05261903 0.05490616 0.13360834
## high -0.83537529 0.92160860 -0.77847650
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.08953054 0.56113425 -1.0101364
## indus 0.07218634 -0.22174594 0.2397393
## chas -0.09531663 -0.05996422 0.1169146
## nox 0.27621683 -0.79385816 -1.3189436
## rm -0.13342457 -0.11478356 -0.2125507
## age 0.22966338 -0.41609631 -0.1325351
## dis -0.02814771 -0.22710599 0.2165613
## rad 3.59273354 1.08337388 -0.2786836
## tax 0.18626877 -0.09533914 0.8325966
## ptratio 0.10340365 -0.04811054 -0.3811377
## black -0.09074088 0.06197170 0.1196124
## lstat 0.21828270 -0.22692557 0.4064369
## medv 0.19223067 -0.42219229 -0.1267976
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9580 0.0322 0.0098model_predictors <- dplyr::select(train, -crime, -crime.1)checking the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=train$crime)5 reading the human data
human2 <-read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt",header = T, sep=",")5.1 Graphical overview of the data
library(ggplot2)
library(GGally)
ggpairs(human2, lower = list(combo = wrap("facethist", bins = 20)))
========================================================================================= Correlations among 8 variables can be seen from the above ggpairs plot. It is noticeble that there exits a highest positive correlation between Life expectancy at birth and Expected years of schooling . Whereas, there exists a highest negative correlation between Maternal mortality ratio and Life expectancy at birth. =========================================================================================
5.1 summaries of the varaibles in the human data
summary(human2)## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## GNI Mat.Mor Ado.Birth Parli.F
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.505.2 Prinicipal component analysis on the non-standardised human data
library(ggfortify)
pca_human2 <- prcomp(human2)
summary(pca_human2)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000
## PC8
## Standard deviation 0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion 1.0000================================================================================== We can obtain 8 principal components PC1-8. Each of these explains a percentage of the total variation in the dataset. That is to say: PC1 explains 99% of the total variance, which means that nearly all of the information in the dataset (8 variables) can be encapsulated by just that one Principal Component. PC2 explains 0.001 of the variance. So, by knowing the position of a sample in relation to just PC1 and PC2, we can get a very accurate view on where it stands in relation to other samples, as just PC1 and PC2 can explain 99% of the variance. ==================================================================================
5.2 Biplot displying the observations by the first two principal components
pca_human2 <- prcomp(human2)
biplot(pca_human2, choices = 1:2, cex=c(0.8,1), col=c("grey40", "deeppink2"))## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
5.3 Standardizing the variables in the human data
human_std <- scale(human2)
summary(human_std)## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :-2.8189 Min. :-2.6247 Min. :-2.7378 Min. :-2.7188
## 1st Qu.:-0.5233 1st Qu.:-0.5484 1st Qu.:-0.6782 1st Qu.:-0.6425
## Median : 0.3503 Median : 0.2316 Median : 0.1140 Median : 0.3056
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5958 3rd Qu.: 0.7350 3rd Qu.: 0.7126 3rd Qu.: 0.6717
## Max. : 2.6646 Max. : 1.6632 Max. : 2.4730 Max. : 1.4218
## GNI Mat.Mor Ado.Birth Parli.F
## Min. :-0.9193 Min. :-0.6992 Min. :-1.1325 Min. :-1.8203
## 1st Qu.:-0.7243 1st Qu.:-0.6496 1st Qu.:-0.8394 1st Qu.:-0.7409
## Median :-0.3013 Median :-0.4726 Median :-0.3298 Median :-0.1403
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.1932 3rd Qu.: 0.6030 3rd Qu.: 0.6127
## Max. : 5.6890 Max. : 4.4899 Max. : 3.8344 Max. : 3.1850=================================================== By scaling the human data all the mean values for the varaibles were reduced to zero ===================================================
5.3 Biplot displying the observations by the first two principal components in the standardized data
pca_human <- prcomp(human_std)
biplot(pca_human, choices = 1:2, cex=c(0.8,1), col=c("grey40", "deeppink2"))
5.4 Biplot displying the observations by the first two principal components in the standardized data
s <- summary(pca_human)
s## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion 0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
## PC7 PC8
## Standard deviation 0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion 0.98702 1.00000pca_pr <- round(100*s$importance[2,], digits = 1)
pca_pr## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
## 53.6 16.2 9.6 7.6 5.5 3.6 2.6 1.3pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)")
biplot(pca_human, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2])
===================================================================================================== Standardised data gives a better idea of the variability, with Maternal mortality ratio and Adolescent birth rate cvontributing to the PC1 which explains upto 53.6 variation. While, F/M ratio in the labour force and Percetange of female representatives in parliament contributing to PC2 which explains upto 16.2% variation. Additionally, Maternal mortality ratio and Adolescent birth rate seem to be strongly correlated, whereas F/M ratio in the labour force and Percetange of female representatives in parliament seem to correlate positively. =====================================================================================================
5.5 Tea data analysis
library(FactoMineR)
library(dplyr)
tea<-read.table("http://factominer.free.fr/factomethods/datasets/tea.txt",header = T, sep="\t")
str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "+60","15-24",..: 4 5 5 2 5 2 4 4 4 4 ...
## $ frequency : Factor w/ 4 levels "+2/day","1 to 2/week",..: 3 3 1 3 1 3 4 2 1 1 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...dim(tea)## [1] 300 36glimpse(tea)## Observations: 300
## Variables: 36
## $ breakfast <fct> breakfast, breakfast, Not.breakfast, Not.brea...
## $ tea.time <fct> Not.tea time, Not.tea time, tea time, Not.tea...
## $ evening <fct> Not.evening, Not.evening, evening, Not.evenin...
## $ lunch <fct> Not.lunch, Not.lunch, Not.lunch, Not.lunch, N...
## $ dinner <fct> Not.dinner, Not.dinner, dinner, dinner, Not.d...
## $ always <fct> Not.always, Not.always, Not.always, Not.alway...
## $ home <fct> home, home, home, home, home, home, home, hom...
## $ work <fct> Not.work, Not.work, work, Not.work, Not.work,...
## $ tearoom <fct> Not.tearoom, Not.tearoom, Not.tearoom, Not.te...
## $ friends <fct> Not.friends, Not.friends, friends, Not.friend...
## $ resto <fct> Not.resto, Not.resto, resto, Not.resto, Not.r...
## $ pub <fct> Not.pub, Not.pub, Not.pub, Not.pub, Not.pub, ...
## $ Tea <fct> black, black, Earl Grey, Earl Grey, Earl Grey...
## $ How <fct> alone, milk, alone, alone, alone, alone, alon...
## $ sugar <fct> sugar, No.sugar, No.sugar, sugar, No.sugar, N...
## $ how <fct> tea bag, tea bag, tea bag, tea bag, tea bag, ...
## $ where <fct> chain store, chain store, chain store, chain ...
## $ price <fct> p_unknown, p_variable, p_variable, p_variable...
## $ age <int> 39, 45, 47, 23, 48, 21, 37, 36, 40, 37, 32, 3...
## $ sex <fct> M, F, F, M, M, M, M, F, M, M, M, M, M, M, M, ...
## $ SPC <fct> middle, middle, other worker, student, employ...
## $ Sport <fct> sportsman, sportsman, sportsman, Not.sportsma...
## $ age_Q <fct> 35-44, 45-59, 45-59, 15-24, 45-59, 15-24, 35-...
## $ frequency <fct> 1/day, 1/day, +2/day, 1/day, +2/day, 1/day, 3...
## $ escape.exoticism <fct> Not.escape-exoticism, escape-exoticism, Not.e...
## $ spirituality <fct> Not.spirituality, Not.spirituality, Not.spiri...
## $ healthy <fct> healthy, healthy, healthy, healthy, Not.healt...
## $ diuretic <fct> Not.diuretic, diuretic, diuretic, Not.diureti...
## $ friendliness <fct> Not.friendliness, Not.friendliness, friendlin...
## $ iron.absorption <fct> Not.iron absorption, Not.iron absorption, Not...
## $ feminine <fct> Not.feminine, Not.feminine, Not.feminine, Not...
## $ sophisticated <fct> Not.sophisticated, Not.sophisticated, Not.sop...
## $ slimming <fct> No.slimming, No.slimming, No.slimming, No.sli...
## $ exciting <fct> No.exciting, exciting, No.exciting, No.exciti...
## $ relaxing <fct> No.relaxing, No.relaxing, relaxing, relaxing,...
## $ effect.on.health <fct> No.effect on health, No.effect on health, No....================================================================================================================================================== Tea dataset contains 300 observations (8)tea consumers) of 36 variables of answeres to a survey about their consumption of tea. These include how they consume tea, how they think of tea and descriptive questions (sex, age, socio-professional category and sport practise). Except for the age, all the variables are categorical. For the age, the data set has two different variables: a continuous and a categorical one ==================================================================================================================================================
library(tidyverse)## -- Attaching packages ----------------------------------------------------------------------------------------------------- tidyverse 1.2.1 --## v tibble 2.1.3 v purrr 0.3.3
## v tidyr 1.0.0 v stringr 1.4.0
## v readr 1.3.1 v forcats 0.4.0## -- Conflicts -------------------------------------------------------------------------------------------------------- tidyverse_conflicts() --
## x plotly::filter() masks dplyr::filter(), stats::filter()
## x dplyr::lag() masks stats::lag()
## x plotly::select() masks MASS::select(), dplyr::select()keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
tea_time <- select(tea, one_of(keep_columns))
summary(tea_time)## Tea How how sugar
## black : 74 alone:195 tea bag :170 No.sugar:155
## Earl Grey:193 lemon: 33 tea bag+unpackaged: 94 sugar :145
## green : 33 milk : 63 unpackaged : 36
## other: 9
## where lunch
## chain store :192 lunch : 44
## chain store+tea shop: 78 Not.lunch:256
## tea shop : 30
## str(tea_time)## 'data.frame': 300 obs. of 6 variables:
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))## Warning: attributes are not identical across measure variables;
## they will be dropped
========================= Tea data is modified to include 300 observations of 6 variables =========================
5.5 Multiple correspondence analysis on the tea data
mca <- MCA(tea_time, graph = FALSE)
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |================================================================================================================================= Dimension 1 seems to explain the highest variance as revealed by MCA analysis. =================================================================================================================================
5.5 Plotting MCA results
plot(mca, invisible=c("ind"), habillage = "quali")
=================================================================== In MCA plot the distance between variable categories gives a measure of their similarity. It is apparent from the plot that for ex, tea bag and chain store are more similar and Other variable is different from all the categories ===================================================================
5.5 Screeplot of MCA analysis
library(factoextra)## Welcome! Related Books: `Practical Guide To Cluster Analysis in R` at https://goo.gl/13EFCZget_eig(mca)## eigenvalue variance.percent cumulative.variance.percent
## Dim.1 0.27937118 15.238428 15.23843
## Dim.2 0.26092645 14.232352 29.47078
## Dim.3 0.21933575 11.963768 41.43455
## Dim.4 0.18943794 10.332978 51.76753
## Dim.5 0.17722310 9.666715 61.43424
## Dim.6 0.15617745 8.518770 69.95301
## Dim.7 0.14375727 7.841306 77.79432
## Dim.8 0.14126310 7.705260 85.49958
## Dim.9 0.11717818 6.391537 91.89111
## Dim.10 0.08660997 4.724180 96.61529
## Dim.11 0.06205294 3.384706 100.00000fviz_screeplot(mca,addlabels = TRUE, ylim = c(0, 50))
====================================================================================== Dimension 1 can explain upto 15.2% variance as can be seen in the Scree plot ======================================================================================
PART I
6.1.1 Reading RATS data
RATS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", header = TRUE, sep = '\t')
library(dplyr)
library(tidyr)
library(ggplot2)
glimpse(RATS)## Observations: 16
## Variables: 13
## $ ID <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <int> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1 <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 5...
## $ WD8 <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 5...
## $ WD15 <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 5...
## $ WD22 <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 5...
## $ WD29 <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 5...
## $ WD36 <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 5...
## $ WD43 <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 5...
## $ WD44 <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 5...
## $ WD50 <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 6...
## $ WD57 <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 6...
## $ WD64 <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 6...summary(RATS)## ID Group WD1 WD8
## Min. : 1.00 Min. :1.00 Min. :225.0 Min. :230.0
## 1st Qu.: 4.75 1st Qu.:1.00 1st Qu.:252.5 1st Qu.:255.0
## Median : 8.50 Median :1.50 Median :340.0 Median :345.0
## Mean : 8.50 Mean :1.75 Mean :365.9 Mean :369.1
## 3rd Qu.:12.25 3rd Qu.:2.25 3rd Qu.:480.0 3rd Qu.:476.2
## Max. :16.00 Max. :3.00 Max. :555.0 Max. :560.0
## WD15 WD22 WD29 WD36
## Min. :230.0 Min. :232.0 Min. :240.0 Min. :240.0
## 1st Qu.:255.0 1st Qu.:267.2 1st Qu.:268.8 1st Qu.:267.2
## Median :347.5 Median :351.5 Median :356.5 Median :360.0
## Mean :372.5 Mean :379.2 Mean :383.9 Mean :387.0
## 3rd Qu.:486.2 3rd Qu.:492.5 3rd Qu.:497.8 3rd Qu.:504.2
## Max. :565.0 Max. :580.0 Max. :590.0 Max. :597.0
## WD43 WD44 WD50 WD57
## Min. :243.0 Min. :244.0 Min. :238.0 Min. :247.0
## 1st Qu.:269.2 1st Qu.:270.0 1st Qu.:273.8 1st Qu.:273.8
## Median :360.0 Median :362.0 Median :370.0 Median :373.5
## Mean :386.0 Mean :388.3 Mean :394.6 Mean :398.6
## 3rd Qu.:501.0 3rd Qu.:510.5 3rd Qu.:516.0 3rd Qu.:524.5
## Max. :595.0 Max. :595.0 Max. :612.0 Max. :618.0
## WD64
## Min. :245.0
## 1st Qu.:278.0
## Median :378.0
## Mean :404.1
## 3rd Qu.:530.8
## Max. :628.06.1.2 Factoring the categorical variables in RATS data
RATS$ID <- factor(RATS$ID)
RATS$Group <- factor(RATS$Group)
glimpse(RATS)## Observations: 16
## Variables: 13
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1 <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 5...
## $ WD8 <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 5...
## $ WD15 <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 5...
## $ WD22 <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 5...
## $ WD29 <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 5...
## $ WD36 <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 5...
## $ WD43 <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 5...
## $ WD44 <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 5...
## $ WD50 <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 6...
## $ WD57 <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 6...
## $ WD64 <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 6...6.1.3 Converting wide form to long form of RATS data
RATSL <- RATS %>%
gather(key = WD, value = Weight, -ID, -Group) %>%
mutate(Time = as.integer(substr(WD,3,4)))
glimpse(RATSL)## Observations: 176
## Variables: 5
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1",...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...summary(RATSL)## ID Group WD Weight Time
## 1 : 11 1:88 Length:176 Min. :225.0 Min. : 1.00
## 2 : 11 2:44 Class :character 1st Qu.:267.0 1st Qu.:15.00
## 3 : 11 3:44 Mode :character Median :344.5 Median :36.00
## 4 : 11 Mean :384.5 Mean :33.55
## 5 : 11 3rd Qu.:511.2 3rd Qu.:50.00
## 6 : 11 Max. :628.0 Max. :64.00
## (Other):110================================================================================================= RATS data contain results from a nutrition study conducted in three groups of rats. The groups were put on different diets, and each animal’s body weight (grams) was recorded repeatedly (approximately) weekly, except in week seven when two recordings were taken) over a 9-week period. Original data format in its wide format contains 16 observations of 13 variables namely “ID” “Group” “WD1” “WD8” “WD15” “WD22” “WD29” “WD36” “WD43” “WD44” “WD50” “WD57” “WD64”, with each rat’s repeated weight measurements in a single row and weekdays in columns.In the long form the data now shows 176 observations with 5 variables ID,Group, WD, Weight and time, Each row represent one time point per rat, so each rat will have data in multiple rows. =================================================================================================
6.1.2 Plotting the RATSL data
ggplot(RATSL, aes(x = Time, y = Weight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:16, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(RATSL$Weight), max(RATSL$Weight)))
6.1.5 Plotting the RATSL data
ggplot(RATSL, aes(x = Time, y = Weight, group = ID)) +
geom_line(aes(linetype = Group))+
scale_x_continuous(name = "Time (days)", breaks = seq(0, 60, 10))+
scale_y_continuous(name = "Weight (grams)")+
theme(legend.position = "top")
======================================================================================= It is obvious from the graphs that the Rats from group 1 have lower weights throughout the study period with minor increase. Rats from group 2 and 3 who have higher weights at the beginning seem to gain weight over the 9 weeks of the study.The phenomena known as tracking. Also it is apparent that there exists substantial intra-group variation in group 2 and 3 ======================================================================================= # 6.1.6 Standardise the variable in RATSL data
RATSL <- RATSL %>%
group_by(Time) %>%
mutate(stdWeight = (Weight - mean(Weight))/sd(Weight) ) %>%
ungroup()
glimpse(RATSL)## Observations: 176
## Variables: 6
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1...
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 44...
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8...
## $ stdWeight <dbl> -1.0011429, -1.1203857, -0.9613953, -0.8421525, -0.8...6.1.7 Plotting again with the standardised Weight
ggplot(RATSL, aes(x = Time, y = stdWeight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:16, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
scale_y_continuous(name = "standardized Weight")
================================================================================================ The tracking phenomena of weight gain across the groups can be seen clearly with a plot of the stanardized values of each observation, suggesting group 2 and 3 seems to gain weight during the study period ================================================================================================
6.1.8 Number of WD, Summary data with mean and standard error of Weight by group and time
n <- RATSL$Time %>% unique() %>% length()
RATSS <- RATSL %>%
group_by(Group, Time) %>%
summarise( mean = mean(Weight), se = sd(Weight)/sqrt(n) ) %>%
ungroup()
glimpse(RATSS)## Observations: 33
## Variables: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,...
## $ Time <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, ...
## $ mean <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 26...
## $ se <dbl> 4.589478, 3.947710, 3.460116, 4.100800, 3.333956, 3.5529...6.1.9 Plotting the mean profiles of RATSS data
ggplot(RATSS, aes(x = Time, y = mean, linetype = Group, shape = Group)) +
geom_line() +
scale_linetype_manual(values = c(1,2,3)) +
geom_point(size=3) +
scale_shape_manual(values = c(1,2,3)) +
geom_errorbar(aes(ymin = mean - se, ymax = mean + se, linetype="1"), width=0.3) +
theme(legend.position = c(0.8,0.8)) +
scale_y_continuous(name = "mean(Weight) +/- se(Weight)")
=========================================================================================================== Mean weight profiles of the 3 different groups can be seen along with indication of varation of the weight at each time point during study period. It is apparent that group 2 and 3 differ considerably compared to group1 both in terms of baseline weight and trend in weight gain ===========================================================================================================
6.1.10 Creating a summary data by Group and ID with mean as the summary variable (ignoring baseline WD 0).
RATSL8S <- RATSL %>%
filter(WD > 0) %>%
group_by(Group, ID) %>%
summarise( mean=mean(Weight) ) %>%
ungroup()
glimpse(RATSL8S)## Observations: 16
## Variables: 3
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ mean <dbl> 261.0909, 237.6364, 260.1818, 266.5455, 269.4545, 274.72...6.1.11 Drawing a boxplot of the mean weights versus Group
ggplot(RATSL8S, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(Weight), Time 1-64")
========================================================================================================= Summary measure method was applied by transforming the repeated measurements into a single value that captures some essential feature od the weight gain over time of a rat in each group and taking all time points except the baseline. The diagram indicates that the mean summary measure is more variable in all groups with an outlier in each group =========================================================================================================
6.1.12 Creating a new RATS data by filtering the outlier
RATSL8S1 <- RATSL8S %>%
filter((mean < 300 & mean > 250 & Group ==1)
|(mean < 550 & Group ==2)
|(mean > 500 & Group ==3))6.1.13 Adjusting the ggplot code and drawing the plot again with the new data
ggplot(RATSL8S1, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(Weight), Time 1-64")
========================================================================================================== An outlier in each group was removed and the mean weight profiles were plotted again. Without outliers, the mean of the weight in group 2 and 3 seems higher than group 1. There seems to be an evidence of difference in location of the summary measure distributions in each group ==========================================================================================================
6.1.14 Adding the baseline from the original data as a new variable to the summary data
RATSL8S2 <- RATSL8S %>%
mutate(baseline = RATS$WD1)6.1.15 Fitting the linear model with the mean as the response
fit <- lm(mean ~ baseline + Group , data = RATSL8S2)6.1.16 Computing the analysis of variance table for the fitted model with anova()
anova(fit)## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## baseline 1 252125 252125 2237.0655 5.217e-15 ***
## Group 2 726 363 3.2219 0.07586 .
## Residuals 12 1352 113
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1=========================================================================================================== By incorporating the baseline values in to the summary measures data and performing ANOVA analysis, we can see that the baseline weight is strongly related to the weights observed after the intervention, but there seems to be no evidence of Group 2 and 3 gaining significantly more weight compared to Group 1 ===========================================================================================================
PART II
6.2.1 Reading the BPRS data
BPRS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep =" ", header = T)
names(BPRS)## [1] "treatment" "subject" "week0" "week1" "week2"
## [6] "week3" "week4" "week5" "week6" "week7"
## [11] "week8"str(BPRS)## 'data.frame': 40 obs. of 11 variables:
## $ treatment: int 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : int 1 2 3 4 5 6 7 8 9 10 ...
## $ week0 : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week1 : int 36 68 55 77 75 43 61 36 43 51 ...
## $ week2 : int 36 61 41 49 72 41 47 38 39 51 ...
## $ week3 : int 43 55 38 54 65 38 30 38 35 55 ...
## $ week4 : int 41 43 43 56 50 36 27 31 28 53 ...
## $ week5 : int 40 34 28 50 39 29 40 26 22 43 ...
## $ week6 : int 38 28 29 47 32 33 30 26 20 43 ...
## $ week7 : int 47 28 25 42 38 27 31 25 23 39 ...
## $ week8 : int 51 28 24 46 32 25 31 24 21 32 ...summary(BPRS)## treatment subject week0 week1
## Min. :1.0 Min. : 1.00 Min. :24.00 Min. :23.00
## 1st Qu.:1.0 1st Qu.: 5.75 1st Qu.:38.00 1st Qu.:35.00
## Median :1.5 Median :10.50 Median :46.00 Median :41.00
## Mean :1.5 Mean :10.50 Mean :48.00 Mean :46.33
## 3rd Qu.:2.0 3rd Qu.:15.25 3rd Qu.:58.25 3rd Qu.:54.25
## Max. :2.0 Max. :20.00 Max. :78.00 Max. :95.00
## week2 week3 week4 week5
## Min. :26.0 Min. :24.00 Min. :20.00 Min. :20.00
## 1st Qu.:32.0 1st Qu.:29.75 1st Qu.:28.00 1st Qu.:26.00
## Median :38.0 Median :36.50 Median :34.50 Median :30.50
## Mean :41.7 Mean :39.15 Mean :36.35 Mean :32.55
## 3rd Qu.:49.0 3rd Qu.:44.50 3rd Qu.:43.00 3rd Qu.:38.00
## Max. :75.0 Max. :76.00 Max. :66.00 Max. :64.00
## week6 week7 week8
## Min. :19.00 Min. :18.0 Min. :20.00
## 1st Qu.:22.75 1st Qu.:23.0 1st Qu.:22.75
## Median :28.50 Median :30.0 Median :28.00
## Mean :31.23 Mean :32.2 Mean :31.43
## 3rd Qu.:37.00 3rd Qu.:38.0 3rd Qu.:35.25
## Max. :64.00 Max. :62.0 Max. :75.00=========================================================== BPRS data contains 40 male subjects were randomly assigned to one of two treatment groups and each subject was rated on the brief psychiatric rating scale (BPRS) measured before treatment began (week 0) and then at weekly intervals for eight weeks. The BPRS assesses the level of 18 symptom constructs such as hostility, suspiciousness, hallucinations and grandiosity; each of these is rated from one (not present) to seven (extremely severe). The scale is used to evaluate patients suspected of having schizophrenia. ============================================================
6.2.2 Factoring the categorical variables in BPRS data
BPRS$treatment <- factor(BPRS$treatment)
BPRS$subject <- factor(BPRS$subject)6.2.3 Converting wide form to long form of BPRS data
BPRSL <- BPRS %>% gather(key = weeks, value = bprs, -treatment, -subject)
BPRSL <- BPRSL %>% mutate(week = as.integer(substr(weeks,5,5)))
glimpse(BPRSL)## Observations: 360
## Variables: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...============================================================= The BPRS data in its original wide form contains In the wide form of BPRS data, we have 11 variables “treatment” “subject” “week0” “week1” “week2” “week3” “week4” “week5” “week6” “week7” “week8”. In wide from a subject’s repeated responses are in a single row and each response is in separate column. In the long form the variables are “treatment” “subject” “weeks” “bprs” “week” “stdbprs” . It now contains 360 observations and 6 variables. In the long form of the data we have in each row representing one time point per subject. So each subject will have data in multiple rows. Any variable that do not change aross time will have the same value ib all rows =============================================================
6.2.4 Plotting the BPRSL data
ggplot(BPRSL, aes(x = week, y = bprs, linetype = subject)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(BPRSL$bprs), max(BPRSL$bprs)))
============================================================================================== By taking into account the longitudinal structure of the BRPS data by joining together the points belonging to each suject to the bprs profiles of individual subject, we can see that there exists substantial variation between individuals and within individuals acroo time ==============================================================================================
6.2.5 Creating a regression model BPRS_reg
BPRS_reg <- lm(bprs ~ week + treatment, data = BPRSL)
summary(BPRS_reg)##
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRSL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16================================================== We are fitting a multiple regression model with bprs as response and week and treatment as explanatory variables. It can be seen that treatment 1 and 2 dont seem to differ conditional on time ==================================================
6.2.6 Creating a random intercept model for BPRSL data
library(lme4)## Loading required package: Matrix##
## Attaching package: 'Matrix'## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpackBPRS_ref <- lmer(bprs ~ week + treatment + (1 | subject), data = BPRSL, REML = FALSE)
print(BPRS_ref)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: BPRSL
## AIC BIC logLik deviance df.resid
## 2748.712 2768.143 -1369.356 2738.712 355
## Random effects:
## Groups Name Std.Dev.
## subject (Intercept) 6.885
## Residual 10.208
## Number of obs: 360, groups: subject, 20
## Fixed Effects:
## (Intercept) week treatment2
## 46.4539 -2.2704 0.5722======================================================== We can fit the randrom intercept model for the same two explanatory variables: week and treatment. Fitting a random intercept model allows the linear regression fit for each subject to differ in intercept from other subjects. ========================================================
6.2.7 creating a random intercept and random slope model for BPRSL data
BPRS_ref1 <- lmer(bprs ~ week + treatment + (week | subject), data = BPRSL, REML = FALSE)
print(BPRS_ref1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
## Data: BPRSL
## AIC BIC logLik deviance df.resid
## 2745.440 2772.643 -1365.720 2731.440 353
## Random effects:
## Groups Name Std.Dev. Corr
## subject (Intercept) 8.0512
## week 0.9803 -0.51
## Residual 9.8707
## Number of obs: 360, groups: subject, 20
## Fixed Effects:
## (Intercept) week treatment2
## 46.4539 -2.2704 0.5722======================================================== We can also fit the ransom intercept and random slope model to the bprs data. Fitting a random intercept and random slope model allows the linear regression firs for each individual to differ in intercept but also in slope. It is possible to account for the individual difference in the subjects’ bprs profiles,but also the effect of time ========================================================
6.2.8 Performing an ANOVA test on the two models of BPRS data
anova(BPRS_ref1, BPRS_ref)## Data: BPRSL
## Models:
## BPRS_ref: bprs ~ week + treatment + (1 | subject)
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## BPRS_ref 5 2748.7 2768.1 -1369.4 2738.7
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1============================================================================= BPRS_ref1 seem to fit better according to significance level of BPRS_fit1 and BPRS-fit ratio =============================================================================
6.2.9 Creating a random intercept and random slope model
BPRS_ref2 <- lmer(bprs ~ week * treatment + (week | subject), data = BPRSL, REML = FALSE)
summary(BPRS_ref2)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week * treatment + (week | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2744.3 2775.4 -1364.1 2728.3 352
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0512 -0.6271 -0.0768 0.5288 3.9260
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.9964 8.0620
## week 0.9687 0.9842 -0.51
## Residual 96.4707 9.8220
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.8856 2.2521 21.262
## week -2.6283 0.3589 -7.323
## treatment2 -2.2911 1.9090 -1.200
## week:treatment2 0.7158 0.4010 1.785
##
## Correlation of Fixed Effects:
## (Intr) week trtmn2
## week -0.650
## treatment2 -0.424 0.469
## wek:trtmnt2 0.356 -0.559 -0.840========================================================== With random intercept and slope model fit we can assess interaction between treatment and week. It looks like interaction model is not a better fir for bprs data ==========================================================
6.2.10 Perform an ANOVA test on the two models
anova(BPRS_ref2, BPRS_ref1)## Data: BPRSL
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref2: bprs ~ week * treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4
## BPRS_ref2 8 2744.3 2775.4 -1364.1 2728.3 3.1712 1 0.07495 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 16.2.11 Drawing the plot of BPRSL
ggplot(BPRSL, aes(x = week, y = bprs, treatment = subject)) +
geom_line(aes(linetype = treatment))+
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_x_continuous(name = "Time (week)")+
scale_y_continuous(name = "Observed bprs (psychiatric rating scale)") +
theme(legend.position = "top")
6.2.12 Creating a vector of the fitted values
Fitted <- fitted(BPRS_ref2)6.2.13 Creating a new column fitted to BPRSL data
BPRSL <- BPRSL %>%
mutate(Fitted)6.2.14 Drawing the plot of BPRSL
ggplot(BPRSL, aes(x = week, y = bprs, treatment = subject)) +
geom_line(aes(linetype = treatment))+
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
scale_x_continuous(name = "Time (week)")+
scale_y_continuous(name = "Fitted bprs (psychiatric rating scale)") +
theme(legend.position = "top")
====================================================================================== The interaction model fits well to the observed bprs data ====================================================================================== ***